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2010 | 8 | 5 | 890-907
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Korovkin-type convergence results for non-positive operators

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Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. In this work we present qualitative Korovkin-type convergence results for a class of sequences of non-positive operators, more precisely regular operators with vanishing negative parts under a limiting process. Sequences of that type are called sequences of almost positive linear operators and have not been studied before in the context of Korovkin-type approximation theory. As an example we show that operators related to the multivariate scattered data interpolation technique moving least squares interpolation originally due to Lancaster and Šalkauskas [Surfaces generated by moving least squares methods, Math. Comp., 1981, 37, 141–158] give rise to such sequences. This work also generalizes Korovkin-type results regarding Shepard interpolation [Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38, 170–176] due to the author. Moreover, this work establishes connections and differences between the concepts of sequences of almost positive linear operators and sequences of quasi-positive or convexity-monotone linear operators introduced and studied by Campiti in [Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo (2) Suppl., 1993, 33, 229–238].
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Bibliografia
  • [1] Agratini O., On approximation of functions by positive linear operators, Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacąau, 2006, 16 Suppl., 17–28
  • [2] Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications, de Gruyter Stud. Math., 17, de Gruyter, Berlin-New York, 1994
  • [3] Bohman H., On approximation of continuous and of analytic functions, Ark. Mat., 1952, 2(1), 43–56 http://dx.doi.org/10.1007/BF02591381
  • [4] Campiti M., Convexity-monotone operators in Korovkin theory, In: Proceedings of the 2nd International Conference in Functional Analysis and Approximation Theory, Acquafredda di Maratea (Potenza), September 14–19, 1992, Rend. Circ. Mat. Palermo (2) Suppl., 1994, 33, 229–238
  • [5] DeVore R.A., The Approximation of Continuous Functions by Positive Linear Operators, Lecture Notes in Math., 293, Springer, Berlin-Heidelberg-New York, 1972
  • [6] Farwig R., Multivariate interpolation of scattered data by moving least squares methods, In: Algorithms for Approximation, Proc. IMA Conf., Shrivenham, July 1985, The Institute of Mathematics and its Applications Conference Series, New Series, 10, Clarendon Press, Oxford, 1987, 193–211
  • [7] Farwig R., Rate of convergence for moving least squares interpolation methods: the univariate case, In: Progress in Approximation Theory, Academic Press, 1991, 313–327
  • [8] Korovkin P.P., On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 1953, 90, 961–964, (in Russian)
  • [9] Korovkin P.P., Linear Operators and Approximation Theory, International Monographs on advanced Mathematics & Physics, Hindustan Publishing Corp., Delhi, 1960
  • [10] Lancaster P., Šalkauskas K., Surfaces generated by moving least squares methods, Math. Comp., 1981, 37(155), 141–158
  • [11] Levin D., The approximation power of moving least-squares, Math. Comp., 1998, 67(224), 1517–1531 http://dx.doi.org/10.1090/S0025-5718-98-00974-0
  • [12] Lorentz G.G., Approximation of Functions, 2nd ed., Chelsea Publishing Company, New York, 1986
  • [13] Muñoz-Delgado F.J., Ramírez-González V., Cárdenas-Morales D., Qualitative Korovkin-type results on conservative approximation, J. Approx. Theory, 1998, 94(1), 144–159 http://dx.doi.org/10.1006/jath.1998.3182
  • [14] Netuzhylov H., Sonar T., Yomsatieankul W., Finite difference operators from moving least squares interpolation, ESAIM, Math. Model. Numer. Anal., 2007, 41(5), 959–974 http://dx.doi.org/10.1051/m2an:2007042
  • [15] Nishishiraho T., Convergence of quasi-positive linear operators, Atti Semin. Mat. Fis. Univ. Modena, 1992, 40(2), 519–526
  • [16] Nowak O., Exakte kleinste Quadrate Interpolierende: Konvergenzresultate vom Korovkin-Typ und Anwendungen im Kontext der numerischen Approximation von Erhaltungsgleichungen, Ph.D. thesis, TU Braunschweig, 2009
  • [17] Nowak O., Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38(2), 170–176
  • [18] Nowak O., High-order convergence of Moving Least Squares Interpolation under regular data distributions, preprint available at http://public.me.com/oliver.nowak/highorder.pdf
  • [19] Nowak O., Sonar T., Upwind-biased finite difference approximations from interpolating moving least squares, preprint available at http://public.me.com/oliver.nowak/upwind.pdf
  • [20] Popoviciu T., Asupra demonstratiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare, Editura Academiei Republicii Populare Române, 1951, 1664–1667
  • [21] Popoviciu T., On the proof of Weierstrass’ theorem using interpolation polynomials (translated by Daniela Kasco), East J. Approx., 1998, 4(1), 107–110
  • [22] Shepard D., A two-dimensional interpolation function for irregularly-spaced data, In: Proceedings of the 23rd ACM National Conference, 1968, 517–524
  • [23] Sonar T., Difference operators from interpolating moving least squares and their deviation from optimality, ESAIM, Math. Model. Numer. Anal., 2005, 39(5), 883–908 http://dx.doi.org/10.1051/m2an:2005039
  • [24] Wendland H., Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., 17, Cambridge University Press, Cambridge, 2005
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bwmeta1.element.doi-10_2478_s11533-010-0058-8
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